In this paper, we formulate and prove a general compactness theorem for harmonic maps of Riemann surfaces using Deligne–Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge with the singular set consisting of only “non-regular” nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to \(S^2\), both energy identity and zero distance bubbling hold.

中文翻译：

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具有规则节点的曲面调和图的紧致性

在本文中，我们使用德利涅-芒福德模空间和曲线族制定并证明了黎曼曲面调和映射的一般紧性定理。主要定理表明，给定一系列复杂曲线上的调和映射序列，存在一系列曲线和子序列，使得域和映射都收敛于仅由“非常规”节点组成的奇异集。这为颈部具有零能量和零长度提供了充分的条件。作为推论，可以证明以下已知事实：如果所有域都微分同胚于\(S^2\)，则能量恒等式和零距离冒泡都成立。